Show that the function u = x^3 - 3xy^2 + 3x satisfies the Laplace equation and determine the corresponding analytic function for u.
Understand the Problem
The question is asking to show that the function u = x^3 - 3xy^2 + 3x satisfies the Laplace equation and to determine the corresponding analytic function for u.
Answer
The analytic function corresponding to \( u \) is: \[ f(z) = (x^3 - 3xy^2 + 3x) + i(3y^3 - 3xy^2) \]
Answer for screen readers
The function ( u = x^3 - 3xy^2 + 3x ) satisfies the Laplace equation, and the corresponding analytic function is: [ f(z) = (x^3 - 3xy^2 + 3x) + i(3y^3 - 3xy^2) ]
Steps to Solve
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Identifying the function The given function is ( u = x^3 - 3xy^2 + 3x ).
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Calculating the second partial derivatives To show that ( u ) satisfies the Laplace equation, we need to calculate the second partial derivatives:
- Compute ( \frac{\partial^2 u}{\partial x^2} ): [ \frac{\partial u}{\partial x} = 3x^2 - 3y^2 + 3 ] [ \frac{\partial^2 u}{\partial x^2} = 6x ]
- Compute ( \frac{\partial^2 u}{\partial y^2} ): [ \frac{\partial u}{\partial y} = -6xy ] [ \frac{\partial^2 u}{\partial y^2} = -6x ]
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Applying the Laplace equation The Laplace equation states that: [ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 ] Substituting the second derivatives we calculated: [ 6x - 6x = 0 ] This confirms that the Laplace equation is satisfied.
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Finding the corresponding analytic function An analytic function ( f(z) ) can be expressed in terms of ( x ) and ( y ) using the complex variable ( z = x + iy ). To find ( f(z) ), we can exploit the structure of ( u ). Using the Cauchy-Riemann equations, we can determine:
- ( v = 3y^3 - 3xy^2 ) Thus, the analytic function is: [ f(z) = u + iv = (x^3 - 3xy^2 + 3x) + i(3y^3 - 3xy^2) ]
The function ( u = x^3 - 3xy^2 + 3x ) satisfies the Laplace equation, and the corresponding analytic function is: [ f(z) = (x^3 - 3xy^2 + 3x) + i(3y^3 - 3xy^2) ]
More Information
The Laplace equation is fundamental in various fields such as physics and engineering, often appearing in potential theory and fluid dynamics. Knowing that a function satisfies this equation implies that it is harmonic, which is crucial in many applications.
Tips
- Forgetting to compute both second derivatives can lead to incorrect verification of the Laplace equation.
- Mixing up the signs or misapplying the chain rule when computing derivatives can also result in errors.
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